Tailoring magnetism in silicon-doped zigzag graphene edges

Recently, the edges of single-layer graphene have been experimentally doped with silicon atoms by means of scanning transmission electron microscopy. In this work, density functional theory is applied to model and characterize a wide range of experimentally inspired silicon doped zigzag-type graphene edges. The thermodynamic stability is assessed and the electronic and magnetic properties of the most relevant edge configurations are unveiled. Importantly, we show that silicon doping of graphene edges can induce a reversion of the spin orientation on the adjacent carbon atoms, leading to novel magnetic properties with possible applications in the field of spintronics.

. Schematic representation of the ZGNR model used in the present work.
In order to ensure the reliability of our calculations, different parameters were tested on undoped zigzag GNR.

A) Kpoints number
As it can be observed in Fig. S2, the convergence of the number of k-points in the GNR growth direction, i. e., nx1x1 was tested. The convergence was achieved at 5x1x1, where the change in energy was smaller than 0.006 eV. Figure S2. Convergence of the energy of ZGNR with the number of k-points.

B) Space between adjacent layers
The adjacent layers in GNR in y and z directions are separated by a vacuum space in order to avoid interactions between two layers. The dimension of this vacuum space was checked in the y direction, by using two different unit cells: 17.27 x 27 x 15 Å (spacing between cells of 11.32 Å) and 17.27 x 35 x 15 Å (spacing between cells of 19.44 Å). The results are shown in Table  S1 for GNR with ferromagnetic (FM) and antiferromagnetic (AFM) coupling between edges. It is clear from these data that the convergence is already achieved in the smallest unit cell.  Table S1. Energy difference between AFM and FM states as a function of the vacuum space between cells and magnetic couplings between edges (FM stands for ferromagnetic coupling and AFM for antiferromagnetic coupling between edges).
2. Optimized structures along with their spin density 2.1 Addition of one or two substitutional Si atoms Figure S3. Spin density on structures S2-S8. For each case the state with AFM (top) and FM coupling between opposite edges are shown. The ground state is shown next to the formation energy of each structure (E form ), and the second most stable magnetic state is given next to the energy difference with respect to such ground state. The values for yellow (α-spin) and blue (β-spin) isosurfaces are 0.005 e/Å 3 . Carbon and silicon atoms are depicted in brown and blue respectively.
2.1 Addition of one or two substitutional Si atoms in an edge with a carbon vacancy Figure S4. Spin density on structures P3-P17. For each case the state with AFM (top) and FM coupling between opposite edges are shown. The ground state is shown next to the formation energy of each structure (E form ), and the second most stable magnetic state is given next to the energy difference with respect to such ground state. The values for yellow (α-spin) and blue (β-spin) isosurfaces are 0.005 e/Å 3 . Carbon and silicon atoms are depicted in brown and blue respectively.
3. Undoped and unpassivated ZGNR. Figure S5. Spin density on unpassivated and undoped ZGNR. On the left the ground state, i. e., AFM state, and on the right the FM state, that lies 0.06 higher in energy. The values for yellow (α-spin) and blue (β-spin) isosurfaces are 0.005 e/Å 3 . Carbon and silicon atoms are depicted in brown and blue respectively.
4. Spin density of different magnetic states of the S1, P1 and P2 structures and the relative energy with respect to the magnetic ground state.
Regarding S1, in the most stable configuration the magnetic coupling between the unpaired electrons on opposite edges is AFM ( Figure S6), as it is in the undoped ZGNR, where Si induces a flip of the electron spin on one of the C atoms located in the next sublattice A position. In addition, other magnetic states are characterized and shown in Figure S6. In the configuration closest in energy (only 30 meV higher) the spin flip effect induced by Si is found too. The difference is that the coupling between edges is FM. The magnetic state where all the carbon atoms are coupled ferromagnetically, with no spin flip, with either AFM and FM coupling between the edges is 90 and 130 meV higher in energy, respectively.
In P1 structure a spin flip on the C atom next to Si is also observed. In the most stable state the coupling between the edges is AFM. The analogous configuration with a FM coupling between edges is 60 meV less stable ( Figure S6). Unlike what is found in S1, in P1 the magnetic state with FM coupling within each edge and AFM between edges is only 40 meV higher in energy. Thus the spin flip induced by Si in S1 is slightly more robust.
In P2 the ground state exhibits a FM magnetic coupling between the edges, where the unpaired electrons of two C atoms on sublattice A are flipped. Within 120 meV from the ground state other magnetic states are found where the electrons of two C atoms are flipped, as well as one state where the coupling within the edge and between the edges is FM ( Figure S6). Figure S6. Spin density on selected S1, P1 and P2 cases. For each case the ground state is given as well as other found magnetic states and their energy difference with respect to the ground state. The values for yellow (α-spin) and blue (β-spin) isosurfaces are 0.005 e/Å 3 . In the most stable state of P1 three consecutive unit cells are shown for the sake of clarity. Carbon and silicon atoms are depicted in brown and blue, respectively. Figure S7. Spin density on selected S1, P1 and P2 ground state cases. The effect of ribbon width and edge passivation is compared. The values for yellow (α-spin) and blue (β-spin) isosurfaces are 0.005 e/Å 3 . Carbon, silicon and hydrogen atoms are depicted in brown, blue and white, respectively.

Effect of ribbon width and edge passivation
6. Spin density of the S1, P1 and P2 structures with silicon atoms replaced by carbon atoms and with frozen geometries. Figure S8. Spin density on selected S1, P1 and P2 cases where Si atoms have been substituted by C and the geometries have been kept fixed. The values for yellow (α-spin) and blue (β-spin) isosurfaces are 0.005 e/Å 3 . 7. C-Si bond distances and the Bader charges on the S1 structure. Figure S9. C-Si bond distances (Å), along with the bader charges (|e -|) around the Si atom on structure S1. The values for yellow (α-spin) and blue (β-spin) isosurfaces are 0.005 e/Å 3 . Carbon and silicon atoms are depicted in brown and blue, respectively.

Spin density on the most stable non-substitutional edge adsorption of Si on ZGNR
The non-substitutional edge adsorption of Si was explored as a reference. The most stable structure of the Si adatom is shown in Figure S7 and consists of a Si atom bonded to two carbon atoms on the edge of the ZGNR. In this case, a spin-flip is observed only for a magnetic configuration lying 0.23 eV higher in energy than the ground state, which shows no inversion at all. Figure S10. Spin density on the most stable non-substitutional edge adsorption of Si on ZGNR. The ground state is given, as well as other stable magnetic states and their energy difference with respect to the ground state. The values for yellow (α-spin) and blue (β-spin) isosurfaces are 0.005 e/Å 3 . Carbon and silicon atoms are depicted in brown and blue, respectively.